Worksheet: Dividing Polynomials - Long & Synthetic Division ... - Free Printable

Problem Analysis:

The task involves solving a series of equations and simplifying expressions. The goal is to solve each equation or simplify each expression step by step, ensuring clarity in the process.

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Given Equations/Expressions:

1. $ \frac{3x^2}{6} = \frac{x^2}{2} $
2. $ \frac{4x^2}{8} = \frac{x^2}{2} $
3. $ 0.5x^2 = \frac{x^2}{2} $
4. $ -\frac{3x^2}{6} + \frac{6x^2}{6} = \frac{3x^2}{6} $
5. $ 3x^2 - 6x^2 + 9x^2 = 6x^2 $
6. $ \frac{1}{2}x^2 - \frac{1}{4}x^2 - \frac{1}{8}x^2 = \frac{1}{8}x^2 $

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Solution:

#### (1) Simplify: $ \frac{3x^2}{6} = \frac{x^2}{2} $
- Start with the left-hand side (LHS): $ \frac{3x^2}{6} $.
- Simplify the fraction: $ \frac{3}{6} = \frac{1}{2} $.
- Therefore, $ \frac{3x^2}{6} = \frac{1}{2}x^2 = \frac{x^2}{2} $.
- The right-hand side (RHS) is already $ \frac{x^2}{2} $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{\frac{x^2}{2}}
$$

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#### (2) Simplify: $ \frac{4x^2}{8} = \frac{x^2}{2} $
- Start with the LHS: $ \frac{4x^2}{8} $.
- Simplify the fraction: $ \frac{4}{8} = \frac{1}{2} $.
- Therefore, $ \frac{4x^2}{8} = \frac{1}{2}x^2 = \frac{x^2}{2} $.
- The RHS is already $ \frac{x^2}{2} $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{\frac{x^2}{2}}
$$

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#### (3) Simplify: $ 0.5x^2 = \frac{x^2}{2} $
- Note that $ 0.5 = \frac{1}{2} $.
- Therefore, $ 0.5x^2 = \frac{1}{2}x^2 = \frac{x^2}{2} $.
- The RHS is already $ \frac{x^2}{2} $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{\frac{x^2}{2}}
$$

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#### (4) Simplify: $ -\frac{3x^2}{6} + \frac{6x^2}{6} = \frac{3x^2}{6} $
- Combine the terms on the LHS:
$$
-\frac{3x^2}{6} + \frac{6x^2}{6} = \frac{-3x^2 + 6x^2}{6} = \frac{3x^2}{6}.
$$
- The RHS is already $ \frac{3x^2}{6} $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{\frac{3x^2}{6}}
$$

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#### (5) Simplify: $ 3x^2 - 6x^2 + 9x^2 = 6x^2 $
- Combine the terms on the LHS:
$$
3x^2 - 6x^2 + 9x^2 = (3 - 6 + 9)x^2 = 6x^2.
$$
- The RHS is already $ 6x^2 $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{6x^2}
$$

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#### (6) Simplify: $ \frac{1}{2}x^2 - \frac{1}{4}x^2 - \frac{1}{8}x^2 = \frac{1}{8}x^2 $
- Combine the terms on the LHS:
$$
\frac{1}{2}x^2 - \frac{1}{4}x^2 - \frac{1}{8}x^2 = x^2 \left( \frac{1}{2} - \frac{1}{4} - \frac{1}{8} \right).
$$
- Find a common denominator for the fractions inside the parentheses:
$$
\frac{1}{2} = \frac{4}{8}, \quad \frac{1}{4} = \frac{2}{8}, \quad \frac{1}{8} = \frac{1}{8}.
$$
- Substitute these into the expression:
$$
\frac{1}{2}x^2 - \frac{1}{4}x^2 - \frac{1}{8}x^2 = x^2 \left( \frac{4}{8} - \frac{2}{8} - \frac{1}{8} \right).
$$
- Simplify the fractions:
$$
\frac{4}{8} - \frac{2}{8} - \frac{1}{8} = \frac{4 - 2 - 1}{8} = \frac{1}{8}.
$$
- Therefore:
$$
\frac{1}{2}x^2 - \frac{1}{4}x^2 - \frac{1}{8}x^2 = \frac{1}{8}x^2.
$$
- The RHS is already $ \frac{1}{8}x^2 $.
- Conclusion: The equation is true as both sides are equal.

$$
\boxed{\frac{1}{8}x^2}
$$

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Final Answers:

1. $ \boxed{\frac{x^2}{2}} $
2. $ \boxed{\frac{x^2}{2}} $
3. $ \boxed{\frac{x^2}{2}} $
4. $ \boxed{\frac{3x^2}{6}} $
5. $ \boxed{6x^2} $
6. $ \boxed{\frac{1}{8}x^2} $